List of top Questions asked in KEAM

\( \int \frac{(x+1)^2}{x^2+1} \, dx = \)

\( \int \frac{(x+1)^2}{x^2+1} \, dx = \)

\( \int \frac{x^{n-1}}{x^n + a^n} dx = \)

\( \int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx = \)

\( \int \frac{4e^x}{2e^x - 5e^{-x}} dx = \)

\( \int \frac{x^5}{\sqrt{1+x^3}} \, dx = \)

The slope of the normal to the curve \( y = x^3 - 3x^2 + 2x + 1 \) at the point where the tangent is horizontal is

A spherical balloon is being inflated such that its radius increases at a constant rate of \( 2 \,\text{cm/sec} \). The rate at which the volume is increasing when the radius is \( 5 \,\text{cm} \) is

If \( y = m\log x + nx^2 + x \) has its extreme values at \( x = 2 \) and \( x = 1 \), then \( 2m + 10n = \)

The function \( f(x) = \sin x - kx - c \), where \( k \) and \( c \) are constants, decreases always when

The chord joining the points \( (5,5) \) and \( (11,227) \) on the curve \( y = 3x^2 - 11x - 15 \) is parallel to tangent at a point on the curve. The abscissa of the point is

The slope of the tangent to the curve \( y = 3x^2 - 5x + 6 \) at \( (1,4) \) is

If a circular plate is heated uniformly, its area expands $3c$ times as fast as its radius, then the value of $c$ when the radius is $6$ units, is

Let \( f(x) = (3\sin^2(10x+11) - 7)^2 \) for \( x \in \mathbb{R} \). Then the maximum value of the function is

If \( y = \frac{x}{x+1} + \frac{x+1}{x} \), then \( \frac{d^2y}{dx^2} \) at \( x=1 \) is equal to

If \( |t|<1 \), \( \sin x = \frac{2t}{1+t^2} \), \( \tan y = \frac{2t}{1-t^2} \), then \( \frac{dy}{dx} \) is

If \( y = \sec(\tan^{-1} x) \), then \( \frac{dy}{dx} \) is equal to

If \( f(x) = 3x^2 - 7x + 5 \), then \( \lim_{x \to 0} \frac{f(x) - f(0)}{x} \) is equal to

If \( y^2 = 100\tan^{-1}x + 45\sec^{-1}x + 100\cot^{-1}x + 45\cosec^{-1}x \), then \( \frac{dy}{dx} \) is

If \( f(x) = \sin^{-1} \left( \frac{1 - \cos 2x}{2 \sin x} \right) \), then \( |f'(x)| \) is equal to

The functions \(f, g\) and \(h\) satisfy the relations \( f'(x)=g(x+1) \) and \( g'(x)=h(x-1) \). Then \( f''(2x) \) is equal to

\[ \lim_{x \to \infty} \left(\frac{x^2}{3x-2} - \frac{x}{3}\right) \]

The value of $\displaystyle \lim_{y \to \infty} \left[ y \sin\left(\frac{1}{y}\right) - \frac{1}{y} \right]$ is equal to}

The number of points at which the function \( f(x)=\frac{1{\log_e|x|} \) is discontinuous is}

\[ \lim_{x \to 0} \left(\frac{10\sin 9x}{9\sin 10x}\right) \left(\frac{8\sin 7x}{7\sin 8x}\right) \left(\frac{6\sin 5x}{5\sin 6x}\right) \left(\frac{4\sin 3x}{3\sin 4x}\right) \left(\frac{\sin x}{\sin 2x}\right) \]